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On nonnegatively curved hypersurfaces in \(\mathbb{H}^{n+1}\). (English) Zbl 1406.53060

In [Proc. Symp. Pure Math. 54, Part 3, 37–44 (1993; Zbl 0804.53083)], S. B. Alexander and R. J. Currier conjectured that, except for covering maps of equidistant surfaces in \(\mathbb{H}^3\), every nonnegatively curved immersed hypersurface in \(\mathbb{H}^{n+1}\) is properly embedded. They also sketched a proof of this conjecture for dimension \(n\geq3\) following a suggestion of Gromov, but the conjecture remained open in the case \(n=2\). In this paper, the authors present a complete proof of this conjecture in higher dimensions (\(n\geq3\)), as well as in the case when \(n=2\).

MSC:

53C40 Global submanifolds
57R40 Embeddings in differential topology

Citations:

Zbl 0804.53083
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Full Text: DOI arXiv

References:

[1] Alexander, S.; Currier, RJ, Nonnegatively curved hypersurfaces of hyperbolic space and subharmonic functions, J. Lond. Math. Soc., 41, 347-360, (1990) · Zbl 0722.53053 · doi:10.1112/jlms/s2-41.2.347
[2] Alexander, S.; Currier, RJ, Hypersurfaces and nonnegative curvature, Proc. Symp. Pure Math., 54, 37-44, (1993) · Zbl 0804.53083 · doi:10.1090/pspum/054.3/1216609
[3] Bonini, V.; Espinar, JM; Qing, J., Correspondences of hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant PDEs, Proc. Am. Math. Soc., 138, 4109-4117, (2010) · Zbl 1202.53053 · doi:10.1090/S0002-9939-2010-10512-9
[4] Bonini, V.; Espinar, JM; Qing, J., Hypersurfaces in hyperbolic space with support function, Adv. Math., 280, 506-548, (2015) · Zbl 1317.53072 · doi:10.1016/j.aim.2015.05.001
[5] Bonini, V., Qing, J., Zhu, J.: Weekly horospherically convex surfaces in hyperbolic 3-space. Ann. Glob. Anal. Geom. (to appear) · Zbl 1385.53043
[6] Bryant, RL, Surfaces of mean curvature one in hyperbolic space, Astérisque, 154-155, 321-347, (1987)
[7] Carron, G.; Herzlich, M., Conformally flat manifolds with nonnegative Ricci curvature, Compos. Math., 142, 798-810, (2006) · Zbl 1100.53034 · doi:10.1112/S0010437X06002016
[8] Cartan, E., Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. (4), 17, 177-191, (1938) · Zbl 0020.06505 · doi:10.1007/BF02410700
[9] Cecil, T.: Lie Sphere Geometry, with Application to submanifolds, 2nd edn. Universitext Springer, New York (2008) · Zbl 1134.53001
[10] Cheeger, J.; Gromoll, D., The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differ. Geom., 6, 119-128, (1971) · Zbl 0223.53033 · doi:10.4310/jdg/1214430220
[11] Chern, SS; Lashof, RK, On the total curvature of immersed manifolds, Mich. Math. J., 5, 5-12, (1958) · Zbl 0095.35803 · doi:10.1307/mmj/1028998005
[12] Currier, RJ, On surfaces of hyperbolic space infinitesimally supported by horospheres, Trans. Am. Math. Soc., 313, 419-431, (1989) · Zbl 0679.53045 · doi:10.1090/S0002-9947-1989-0935532-0
[13] Croke, CB; Karcher, H., Volume of small balls on open manifolds: lower bounds and examples, Trans. Am. Math. Soc., 309, 753-762, (1988) · Zbl 0709.53033 · doi:10.1090/S0002-9947-1988-0961611-7
[14] Do Carmo, MP; Warner, FW, Rigidity and convexity of hypersurfaces in spheres, J. Differ. Geom., 4, 134-144, (1970) · Zbl 0201.23702
[15] Epstein, CL, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math., 372, 96-135, (1986) · Zbl 0591.30018
[16] Epstein, CL, The asymptotic boundary of a surface imbedded in \({\mathbb{H}}^{3}\) with nonnegative curvature, Mich. Math. J., 34, 227-239, (1987) · Zbl 0625.53053 · doi:10.1307/mmj/1029003554
[17] Epstein, C.L.: Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space. Unpublished (1986). http://www.math.upenn.edu/ cle/papers/index.html
[18] Espinar, JM; Gálvez, JA; Mira, P., Hypersurfaces in \({\mathbb{H}}^{n+1}\) and conformally invariant equations: the generalized Christoffel and Nirenberg problems, J. Eur. Math. Soc., 11, 903-939, (2009) · Zbl 1203.53057 · doi:10.4171/JEMS/170
[19] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977) · doi:10.1007/978-3-642-96379-7
[20] Hadamard, J., Sur certaines proprétës des trajactoires en dynamique, J. Math., 3, 331-387, (1897) · JFM 28.0643.01
[21] Huber, A., On subharmonic functions and differential geometry in the large, Comment. Math. Helv., 32, 13-72, (1957) · Zbl 0080.15001 · doi:10.1007/BF02564570
[22] Li, P.; Schoen, R., \(L^{p}\) and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math., 153, 279-301, (1984) · Zbl 0556.31005 · doi:10.1007/BF02392380
[23] Sacksteder, R., On hypersurfaces with no negative sectional curvature, Am. J. Math., 82, 609-630, (1960) · Zbl 0194.22701 · doi:10.2307/2372973
[24] Schlenker, JM, Hypersurfaces in \({\mathbb{H}}^{n}\) and the space of its horospheres, Geom. Funct. Anal., 12, 395-435, (2002) · Zbl 1011.53046 · doi:10.1007/s00039-002-8252-x
[25] Schoen, R.; Yau, ST, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math., 92, 47-71, (1988) · Zbl 0658.53038 · doi:10.1007/BF01393992
[26] Schoen, R., Yau, S.T.: Lectures on Differential Geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology, vol. 1. International Press (1994) · Zbl 0830.53001
[27] Spivak, M.: A comprehensive introduction to differential geometry, vol. IV. Publish or Perish, Houston (1999) · Zbl 1213.53001
[28] Stoker, JJ, Über die Gestalt der positiv gekrümmten offenen Flächen im dreidimensionalen Raume, Compos. Math., 3, 55-89, (1936) · JFM 62.1460.02
[29] Taliaferro, S., On the growth of superharmonic functions near an isolated singularity I, J. Differ. Equ., 158, 28-47, (1999) · Zbl 0939.31005 · doi:10.1016/S0022-0396(99)80017-5
[30] Taliaferro, S., Isolated singularities of nonlinear elliptic inequalities. II. Asymptotic behavior of solutions, Indiana Univ. Math. J., 55, 1791-1811, (2006) · Zbl 1180.35236 · doi:10.1512/iumj.2006.55.2848
[31] Heijenoort, J., On locally convex manifolds, Commun. Pure Appl. Math., 5, 223-242, (1952) · Zbl 0049.12201 · doi:10.1002/cpa.3160050302
[32] Volkov, YA; Vladimirova, SM, Isometric immersions in the Euclidean plane in Lobachevskii space, Math. Zametki, 10, 327-332, (1971) · Zbl 0234.53047
[33] Yau, ST, Some function-theoretic properties of complete Riemannian manfolds and their applications to geometry, Indiana Univ. Math. J., 25, 659-670, (1976) · Zbl 0335.53041 · doi:10.1512/iumj.1976.25.25051
[34] Zhu, S., The classification of complete locally conformally flat manifolds of nonnegative Ricci curvature, Pac. J. Math., 163, 189-199, (1994) · Zbl 0809.53041 · doi:10.2140/pjm.1994.163.189
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